A note on sign-changing solutions to supercritical Yamabe-type equations

On a closed Riemannian manifold $(M^n,g)$, we consider the Yamabe-type equation $-{\mathrm{\Delta }}_{g}u+\lambda u=\lambda \mathrm{\mid }u{\mathrm{\mid }}^{q-1}u$, where $\lambda \in {\mathbb{R}}_{+}$ and $q>1$. We assume that $M$ admits a proper isoparametric function $f$ with focal submanifolds of positive dimension. If $k>0$ is the minimum of the dimensions of the focal submanifolds of $f$, we let $q^{\ast }=\frac{n-k+2}{n-k-2}$. We prove the existence of infinite $f$-invariant sign-changing solutions to the equation when $1<q<q^{\ast }$.